The two major projects included under this heading include the analysis of error in and optimization of NMR experiments, and the development of an exact theory for the calculation of probability densities for use in the analysis of crystallographic data. Together with J.A. Ferretti we have completed the first part of a project on the optimal choice of apodization functions for data reduction in FT-NMR for the purpose of reducing bias and fluctuations insofar as this is possible. While spectrometer manufacturers allow a limited choice of apodization functions we have shown these are not necessarily optimal and have explored the range of parameters required for the specification of these functions. In the area of crystallography we have completed a calculation of the probability density function for the three-phase invariant in space group Pl. This has involved a very complicated use of symmetries to reduce the computer running time in the evaluation of the six-dimensional Fourier representation. Our earlier attempts to evaluate such series without taking symmetries into account proved fruitless even on a supercomputer. The results of our investigation agree with qualitative results obtained by us in earlier calculations related to direct methods of phase determination, that the approximate technique in current use err in being too conservative. This error can be as much as 100% in the presence of atomic heterogeneity. A second project currently being completed relates to the effects of anomalous scatterers on the probability density function for the scattering intensity. The results of our theory show that the major effect of anomalous scattering appears to low intensities. In this case the necessary theory to take account of complex scattering factors required the development of new mathematical techniques for calculating the, probability density. These techniques are now being applied to a variety of other problems in crystallographic analysis whose mathematical structure is similar.